# Full text of "Treatise On Analysis Vol-Ii"

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```396       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

Let Fffc denote the sum of the subspaces T(<pLik)(Rji) for 1 <£y g /, and let Fik denote
the image of Fjfc under the invertible operator

M(wtl)   •-   Af(wljk)\
M(w21)   •••   Af(H>2fc)\

M(wkl)    •••    M(wkk)]

on Gk . Then F* is the sum of the Ftk (1 ^ / ^ &). (Consider the projector PF •)

11. THE SPECTRAL THEORY OF HILBERT

(15.11.1) Let E be a separable Hilbert space and TV a normal continuous
operator on E (15.4.1 1 ). If.fi/is the closure of the (commutative) subalgebra of
&(E) generated by 1E, N and N*, then ((15.4.13) and (15.4.15)) s/ is a
separable star algebra, and the mapping co : ji— >#(7V) is a homeomorphism of
X(cC/) onto Sp(TV) c C (the spectrum of N with respect to J*?(E), that is to say
the spectrum of N as defined in (11.1)). Hence it follows from the Gelfand-
Neumark theorem (15.4.14) that the mapping /W ^ " * (/ o co) is a faithful
representation of ^c(Sp(AO) on E, which can therefore be extended to a
faithful representation of ^c(Sp(AO) on E (15.10.8). We denote this represen-
tation by/W/(AO. This notation is justified by the fact that, if /and g are
any two functions belonging to ^c(Sp(AO), then in the algebra & (E) we have

=/(AO + g(N),       (fg)(N) =
(15.11.1.1)

(15.11.1.2)

(where lSp(JV) is the identity mapping f H» f of Sp(N)). The Gelfand-Neumark
theorem also shows that, for every continuous function / on Sp(JV), we have

(15.11.1.3)                 ||/(N)||=P(/(JV))=   sup |/(C)|

(cf. (15.11.8.1)).

Moreover ((15.10.3) and (15.10.8)), there exists a continuous sesquilinear
mapping (x, y) ^mx>y of E x E into the Banach space Mc(Sp(AT)) of complex
measures on Sp(JV) such that my> x = mx y and

(15.11.2)                     (/(JV) • x | y) =   /(O dm, .(0be identified with LC(^*)» and
```